The first eigenvalue of the Dirac operator on Kähler manifolds |
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Authors: | K. D. Kirchberg |
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Affiliation: | Sektion Mathematik Humboldt-Universität zu Berlin Postfach 1297, 1086, Berlin, Germany |
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Abstract: | If M2m is a closed Kähler spin manifold of positive scalar curvature R, then each eigenvalue λ of type r (r {1, …, [(m + 1)/2]}) of the Dirac operator D satisfies the inequality λ2 ≥ rR0/4r − 2, where R0 is the minimum of R on M2m. Hence, if the complex dimension m is odd (even) we have the estimation for the first eigenvalue of D. In the paper is also considered the limiting case of the given inequalities. In the limiting case with m = 2r − 1 the manifold M2m must be Einstein. The manifolds S2, S2 × S2, S2 × T2, P3(), F(), P3() × T2 and F(3) × T2, where F(3) denotes the flag manifold and T2 the 2-dimensional flat torus, are examples for which the first eigenvalue of the Dirac operator realizes the limiting case of the corresponding inequality. In general, if M2m is an example of odd complex dimension m, then M2m × T2 is an example of even complex dimension m + 1. The limiting case is characterized by the fact that here appear eigenspinors of D2 which are Kählerian twistor-spinors. |
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Keywords: | Kähler spin manifold Dirac operator eigenvalue Einstein manifold |
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